Aircraft: EMB ERJ 170/175 ER/LR
Phase of flight: Climb
Anomaly: Aircraft Equipment Problem Critical

Climbing out of ZZZ and after being handed off to center we were cleared to our filed cruise altitude. Shortly after we heard a very loud and completely random banging noise over the intercom. The MCDU radio page show a red 'micstck' message but it would not clear or reset after the 90 seconds automatic reset. We were unable to transmit over com 1. We tried to reach center over com 2 on assigned frequency and 121.5 but unable to hear anything inbound. Ran the lost communication QRH procedure with no luck on clearing or resetting the radios or com panels. So we entered 7600 in the transponder. There was weather in ZZZ and we had been in IMC flight conditions most of the climbout. I instructed the PF to continue on our filed flight plan as expected; in addition to fly our filed speeds and try to match our scheduled arrival time for the destination airport. I continued to work on re-establishing communications with ATC. As the loud banging noise persisted; I tried to call the cabin and check with the flight attendants to see if an interphone was possibly causing an issue. We were unable to talk to the flight attendants over the cabin interphone so we passed a note over the door notifying them that the interphone was inoperable and to bang on the door and pass a note if they needed our attention.I then proceeded to try and use the com 3 backup radio to reestablish com with ATC. Once again it showed we could transmit but not receive any inbound audio. At that time I switched com 3 back to DATA mode and sent a msg to dispatch; they too had been trying to get a hold of us. We did find ACARS to be operational. I informed dispatch over ACARS that we would plan to continue as filed on time to ZZZ1 and we would be needed to start our decent just south of ZZZ VOR for the filed arrival and transition to the ILS XXL for approach and landing. I also asked them to contact ATC advice of our situation and to coordinate landing clearance lights with ZZZ1 tower. I sent maintenance control a MSG asking if they had any reset procedures for the radios or com panels. Maintenance responded with no solution or procedure while airborne.The FO and I then tried every known way to clear the system; we unplugged all headsets; tried the masks; and cycled all PTT buttons including the jumpseat panel with no luck. We looked for circuit breaker issues; but it appeared the system had just locked up. As we analyze the situation we came to the conclusion that the only communications we had available was our pilot to pilot intercom; with the erratic banging noise and ACARS. We were also somewhat confident we could transmit in the blind on com 2 and did so on 121.5 changing altitudes on the descent and approach. We got back with the flight attendants to find out the cabin PA and FA to FA Interphone were also INOP; we established an emergency communication procedure if needed. We received a descent clearance via ACARS to cross 20 north of PXR at FL250; we complied and transmitted in the blind to advice we were starting down. Another ACARS MSG was received advising ZZZ1 approach would like us to call them on a cell phone when below 10;000 for a landing clearance; and did comply with the cell phone call when work load permitted. Weather was VFR. In ZZZ1 so we looked for green lights for landing and taxi clearance as well. Once at the gate maintenance was unable to clear the situation without first completing a power reset procedure. I have never had a communication failure of the complexity before. The airplane was in an airworthy state before the flight with all systems operating normal up to this event. As a crew we all did our best to correct the situation; but after realizing this was not going to clear while airborne; we all worked together to deal the best we could. The PF and I did our best to follow lost communication procedures as defined by the FAR/AIM. We did our best to not confuse ATC of our intentions to help maintain traffic separation. ERJ-175 Captain reported a communication failure during climbout. ACARS was operational and used to communicate with Dispatch and relay some ATC instructions. A cell phone was used to call Approach Control for landing clearance.